1. Joint Lecture Groningen-Osaka
Spontaneous Breaking of Chiral Symmetry in Hadron Physics
30 Sep 09:00- CEST/16:00- JST Atsushi HOSAKA
07 Oct 09:00- CEST/16:00- JST
Nuclear Structure
21 Oct 09:00- CEST/16:00- JST Nasser KALANTAR-NAYESTANAKI
28 Oct 09:00- CET/17:00- JST
Low-energy tests of the Standard Model
25 Nov 09:00- CET/17:00- JST Rob TIMMERMANS
02 Dec 09:00- CET/17:00- JST
Relativistic chiral mean field model description of finite nuclei
09 Dec 09:00- CET/17:00- JST Hiroshi TOKI
16 Dec 09:00- CET/17:00- JST WRAP-UP/DISCUSSION

2. Spontaneous Breaking of Chiral Symmetry in Hadron Physics
• What does spontaneous mean?
• What is the breaking of Symmetry?
• What is chiral?
• What is hadron? •

3. Contents • General discussions Aspects of symmetry and
of spontaneous breaking
• Concrete examples
NJL model for hadron physics

4. What is symmetric and What is broken symmetry

5. Symmetry
The key concept in the modern Physics
Example of translation

6. Symmetry The key concept in the modern Physics Example of translation
Symmetric
Translation causes
nothing
Uniform density

7. Symmetry The key concept in the modern Physics Example of translation
Symmetric
Translation causes
nothing
Uniform density
Less symmetric

8. Symmetry The key concept in the modern Physics Example of translation
Symmetric
Translation causes
nothing
Uniform density
Less symmetric
Translation changes
the location of the cluster
Localize
Clusterize

9. Symmetry
Example of rotation
Symmetric

10. Symmetry Example of rotation Symmetric Rotation causes nothing
Spherical

11. Symmetry Example of rotation Symmetric Rotation causes nothing
Spherical
Less symmetric

12. Symmetry Example of rotation Symmetric Rotation causes nothing
Spherical
Less symmetric
Rotation changes
the appearance
Deformed

13. Symmetry Example of rotation Symmetric Rotation causes nothing Random
Less symmetric
Rotation changes
the appearacnce
Ordered

14. Spontaneous breaking Symmetric Simple Disordered Less symmetric
Complex
Ordered

15. Spontaneous breaking Phase transition Symmetric Simple Disordered
Symmetry is spontaneously broken
(Dynamical: due to interactions)
Phase transition
Reality in our
world
Less symmetric
Complex
Ordered
With Variety

16. Role of interaction High temperature Kinetic motion > Interaction
Random
Like gas

17. Role of interaction High temperature Kinetic motion > Interaction
Random
Like gas
Interaction breaks the symmetry
=> Spontaneously broken

18. Role of interaction High temperature Kinetic motion > Interaction
Random
Like gas
Interaction breaks the symmetry
=> Spontaneously broken
Low temperature
Kinetic motion < Interaction
Ordered
Like solid

19. Examples of interaction
(1) Translational invariance
H is invariant under
This causes localization (clustering) of a two-particle system
(2) Rotational invariance
This causes deformation of two-particle system (deuteron)

20. Isospin (flavor), chiral, color, ….
(3) Isospin invariance
Iso-spinor
Iso-vector
“Internal symmetry”
Isospin (flavor), chiral, color, ….

21. Recover the broken symmetry
Low T
High T
This does not mean
the phase transition
between them
There is a special way to recover the broken symmetry

22. Recover the broken symmetry
Symmetry transformation
Translation
Rotation p

23. Recover the broken symmetry
Symmetry transformation
Translation
Rotation p
This does not require energy => Zero energy mode
Classical mechanics:
No need to move an object
on a flat/smooth surface
W = Fs = 0
Field theory:
Appearance of a massless particle => pion
m = 0

24. Quantum mechanics
Uncertainty principle

25. Quantum mechanics Starts to move Zeromode excitations
Uncertainty principle p
Starts to move
Uncertainty principle
Flctuations
Zeromode excitations

26. Quantum mechanics Starts to move Zeromode excitations
Uncertainty principle p
Starts to move
Uncertainty principle
Flctuations
Zeromode excitations
For small moment of inertia => Easy to fluctuate
Symmetric states are realized in the quantum world
For large moment of inertia => hard to move
Symmetry is left broken ~ Classical world

27. Collective vs single particle motion

28. Collective vs single particle motion
Nambu-
Goldstone
Boson =
Pion
In these motions, the shape does not change.
The objects move collectively (simultaneously)

29. Collective vs single particle motion
Nambu-
Goldstone
Boson =
Pion
In these motions, the shape does not change.
The objects move collectively (simultaneously)
Massive
Modes=
Mass
generation
Change in the shape requires more energy.
Parts move => Motion of fewer particles

30. Hadrons

31. Where to study? Electromagnetic interaction Strong interaction
Molecule
Many-body dynamics of electrons
around atomic nuclei and/or ions
Atom
Subatomic physics
Strong interaction
Nucleus
Many-body dynamics of nucleons
=> Nuclear Physics mesons
Many-body-dynamics
of quarks and gluson
=> Hadron physics
Nucleons
Mesons
Quarks

32. Where to study? Electromagnetic interaction Subatomic physics
Molecule
Many-body dynamics of electrons
around atomic nuclei and/or ions
Atom
Subatomic physics
Strong interaction
Nucleus
Many-body dynamics of nucleons
=> Nuclear Physics mesons
Many-body-dynamics
of quarks and gluons
Hadron Physics
Nucleons
Mesons
Quarks

33. Atoms Many-electron system => Periodic table Many-electron system
Ne = 1, 2, 3….　[One dimensional plot]
Many-electron system

34. Nuclei
Many-nucleon system (protons and neutrons)

35. Nuclei Many-nucleon system (protons and neutrons) => Nucleat chart
Np = 1, 2, 3….　
Nn = 1, 2, 3…. => [Two-dimensional plot]
Proton number
Neutron number

36. Hadrons Particle Data Many(?)-quark system (u, d, c, s, b, t)
Proton/neutron

37. Particle Data Table
Mesons
Baryons

38. Hadrons Particle Data However Why?
Many(?)-quark system (u, d, c, s, b, t)
Particle Data
Proton/neutron
However
Only qq and qqq?
Why?
Mesons
Baryons

39. Problems of hadron physics
Clay Mathematics Institute, Millennium Problems
Millennium Problems In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the Collège de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem.

40. ７ Yang-Mills Theory => QCD
１Birch and Swinnerton-Dyer Conjecture
２Hodge Conjecture
３Navier-Stokes Equations
４P vs NP
５Poincare Conjecture
６Riemann Hypothesis
７ Yang-Mills Theory => QCD
A. Jaffe and E. Witten
It must have a “mass gap,” that is, there must be some strictly positive constant ∆ such that every excitation of the vacuum has energy at least ∆.
It must have “quark confinement,” that is, even though the theory is described in terms of elementary fields, such as the quarks, that transform non-trivially under S U (3), the physical particle states – such as the proton, neutron, and pion – are S U (3)-invariant.
It must have “chiral symmetry breaking,” which means that the vacuum is potentially invari- ant (in the limit that the quark bare masses vanish) only under a certain subgroup of the full symmetry group that acts on the quark fields.

41. Where qqqq, qqqqq and more ?
Tetraquark
Pentaquark
Exotic hadrons

42. Spontaneous breaking of chiral () symmetry
Yoichiro Nambu

43. Spontaneous breaking of chiral () symmetry
Potential energy surface
of the vacuum
Yoichiro Nambu
Chiral order
parameter
Quarks & gluons
Confinement,
Mass generation
Hadrons & nuclei

44. Dynamics of Spontaneous symmetry breaking in the strongly interacting system

45. Tasks of Physics • Find the ultimate law of everything
• Reconstruct phenomena from the law
They are not independent
due to the presence of interactions
We are on the vacuum.
Particles are the excitations of the vacuum.
Complicated system
Physics is to find the properties of the vacuum and its excitations in the presence of interactions

46. In the microscopic world
A particle
Vacuum = Ground state
is not empty
Particles are interacting with the vacuum
A simply looking system can be more complicated due to the interaction and change its properties drastically.
E.G. from quarks to Hadrons with mass generation

47. Analogy with BCS QED Phonon exchange ee Cooper pair Order parameter
Gauge (local) symmetry
Superconductivity

48. Analogy with BCS QED QCD Phonon exchange ee Strong interaction qq
Cooper pair
Quark-antiquark pair
Order parameter
Gauge (local) symmetry
Superconductivity
Flavor (global) symmetry
Nambu-Goldstone boson

49. • Gap in energy spectrum • Mass of particles
Superconductivity
Hadrons
• Gap in energy spectrum
• Mass of particles
E = 0
Ground state D
Vacuum N N* M
Dirac
mass
Majorana
mass
• Meissner effect
• Exclusion of color electric field
Super
Normal
Super
Normal

50. Chiral symmetry Hand Left Right
Chiral symmetry => Left-hand world has a symmetry (law)
Right-hand world has a symmetry (law)
If they mix, we say that chiral symmetry is broken

51. Massless fermion c c’ = c S-frame S’-frame Right-handed Right-handed
We can not pass the particle moving at the speed of light c
c’ = c
Spin
Chirality remains
unchanged
Spin
S-frame
S’-frame
Right-handed
Right-handed
Right-left do not mixing
Right and left can be independent
Isospin (internal) symmetry can be introduced separately

52. Massive fermion v v’ Right-handed Left-handed
Spin
Spin
Boost can change
from right to left
Right-handed
Left-handed
The word chiral (handedness) comes from this
For massive particle, right and left mix
=> Chiral symmetry is broken

53. Summary 1 Symmetry can be spontaneously broken by interactions.
Symmetry and broken phase can change each other.
(Temperature, density, …)
In the broken phase, symmetry is recovered
by the presence the Nambu-Goldstone mode.
Zero energy mode ~ pion
Collective, and single particle modes are distinguished.
The zero mode (pions) governs the dynamics at low energy.

54. Summary 2 Hadrons are made of quarks and gluons
Baryons qqq, mesons qq*, others (exotics)??
Quark properties changes drastically by the strong interaction
(nearly massless -> massive)
Chiral symmetry is broken spontaneously
Quark masses are dynamically generated (by interaction)
Pions become massless (Nambu-Goldstone mode)

55. Dynamics of L and R <=> V and A
V = R + L, A = R - L
Potential
Vacuum point
Only one
Infinitely many on
-> choose one V A A V
Pions[NG boson] appear

56. Where and how pions appear
Quarks
and gluons
Strong interaction dynamics
Quarks
and mesons
Mass generation
Constituent (quasi) quarks
Pions π π π q q q q q